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ANALYSIS OF SHOCK MOTION IN STBLI USING DNS DATA
Minwei Wu and Pino Martin
Department of Mechanical and Aerospace Engineering, Princeton University,Princeton, NJ 08544, USA
*
Abstract Direct numerical simulation data of a 24◦ com-
pression ramp configuration are used to analyze the shock
motion. The motion can be observed from wall-pressure
and mass-flux signals measured in the free stream. The
characteristic low frequency is in the range of 0.007-0.013
U∞/δ, as found in Wu & Martin (2007). The shock motion
also exhibits high-frequency of O(U∞/δ), small-amplitude
spanwise wrinkling , which is mainly caused by the span-
wise nonuniformity of turbulent structures in the incoming
boundary layer. In studying the low frequency streamwise
oscillation, conditional statistics show that there is no signif-
icant difference in the properties of the incoming boundary
layer when the shock location is upstream or downstream.
The spanwise-mean separation point also undergoes a low
frequency motion and is found to highly correlate with the
shock motion. A small correlation is found between the
low-momentum structures in the incoming boundary layer
and the separation point. Correlations among the spanwise-
mean separation point, reattachment point and the shock
location indicate that the low-frequency shock unsteadiness
is influenced by the downstream flow.
INTRODUCTION
The boundary layer flow over a compression ramp is one
of the canonical shock wave and turbulent boundary layer in-
teraction (STBLI) configurations that have been studied ex-
tensively in experiments since the 1970’s. From this body of
work, we have learned that the shock motion has a frequency
much lower than the characteristic frequency of the incoming
boundary layer. The time scale of the low frequency motion
is O(10δ/U∞−100δ/U∞) as reported in various experiments
such as Dolling & Or (1985), Selig (1988), Dussauge et al.
(2006), and Dupont et al. (2006). In contrast, the character-
istic time scale of the incoming boundary layer is O(δ/U∞).
The scale to normalize the frequency of the shock is still
under debate. However, Dussauge et al. (2006) found that
using StL = fL/U∞, where L is the streamwise length of
the separation bubble, experimental data (covering a wide
range of Mach numbers and Reynolds numbers and vari-
ous configurations) can be grouped between StL = 0.02 and
0.05. Also, the cause of the low frequency motion is still a
research question. Plotkin (1975) proposed a damped spring
model for the shock motion. Andreopoulos & Muck (1987)
concluded that the shock motion is driven by the bursting
events in the incoming boundary layer. However, Thomas
et al. (1994) found no connection between the shock mo-
tion and bursting events in the incoming boundary layer.
Erengil & Dolling (1991) found that there was a correlation
between certain shock motions with pressure fluctuations in
the incoming boundary layer. Beresh et al. (2002) found
that positive velocity fluctuations near the wall correlate
with downstream shock motion. Pirozzoli & Grasso (2006)
analyzed DNS data of a reflected shock interaction and pro-
Table 1: Inflow conditions for the DNS. The Mach number,
Reynolds number based on the momentum thickness, dis-
placement thickness, and boundary layer thickness, bound-
ary layer thickness in wall variables, and skin friction are
given in order of appearance.
M Reθ θ (mm) δ∗ (mm) δ (mm) δ+ Cf
2.9 2300 0.38 1.80 6.4 320 0.0021
posed that a resonance mechanism might be responsible for
the shock unsteadiness. Dussauge et al. (2006) suggested
that the three-dimensional nature of the interaction in the
reflected shock configuration is a key to understanding the
shock unsteadiness. Ganapathisubramani et al. (2006) pro-
posed that very long alternating structures of uniform low-
and high-speed fluid in the logarithmic region of the incom-
ing boundary layer are responsible for the low frequency
motion of the shock. These so called ‘superstructures’ have
been observed in supersonic boundary layers by Samimy et
al. (1994) and are also evident in the elongated wall-pressure
correlation measurements of Owen & Horstmann (1972). Su-
perstructures have also been observed in the atmospheric
boundary layer experiments of Hutchins & Marusic (2007)
and confirmed in DNS of supersonic boundary layers by
Ringuette et al. (2007).
Wu & Martin (2007) presented a direct numerical sim-
ulation of STBLI for a 24◦ compression ramp configuration
at Mach 2.9 and Reynolds number based on momentum
thickness of 2300. They validated the DNS data against
the experiments of Bookey et al. (2005) at matching flow
conditions, and they illustrated the existence of the super-
structures. In this paper, we use the Wu & Martin (2007)
data to analyze the shock unsteadiness. While in previous
experiments the shock motion is usually inferred by measur-
ing the wall pressure, our analyses of the shock motion are
carried mainly in the outer part of the boundary layer and in
the free stream. This is because the Reynolds number that
we consider is much lower than those in typical experiments.
In turn, viscous effects are more prominent, the shock does
not penetrate as deeply as in higher Reynolds number flows,
and the shock location is not well-defined in the lower half
of the boundary layer. In addition, the motion of the sepa-
ration bubble is studied. Table 1 lists the inflow boundary
layer conditions, and Figure 1 shows the computational do-
main and the coordinate system. Notice that we use zn
to denote the wall-normal coordinate and prime symbols to
denote fluctuating quantities. Statistics are gathered over
300δ/U∞ . The characterization of the shock motion and
the unsteadiness of the separation bubble are presented. A
discussion is also presented before the conclusion section.
7δ
9δ4.5δ2.2δ
5δ
y
z
x
Figure 1: Computational domain of the DNS and coordinate
system.
t (δ/U∞)
Pw/P
∞
0 100 200 300
1.0
1.5
2.0
2.5-6.9δ-2.98δ ( i.e. xsep )-2.18δ
(a)
fδ/U∞
Epf
δ/p2 ∞
U∞
10-2 10-1 100 1010
1E-05
2E-05
3E-05
4E-05
5E-05
6E-05
7E-05-6.9δ-2.98δ-2.18δ
(b)
Figure 2: (a) Wall-pressure signals and (b) wall-pressure
energy spectra at different streamwise locations relative to
the ramp corner with y = 1.1δ. From Wu & Martin (2007).
SHOCK MOTION
Figure 2(a) plots three wall-pressure signals measured
at three streamwise locations upstream of the ramp corner
(the corner is located at x = 0) along the spanwise center
line. In the incoming boundary layer at x = −6.9δ, the
normalized magnitude is around one with small fluctuations.
At x = −2.98δ, which is the mean separation point (defined
as the point where the mean skin friction coefficient changes
sign from positive to negative), the magnitude fluctuates
between 1 to 1.2. At x = −2.18δ, the magnitude oscillates
between 1.5 and 2. The corresponding premultiplied energy
spectra are plotted in Figure 2(b). At the mean separation
point, the peak frequency is 0.007U∞/δ. At x = −2.18, the
peak is in 0.01 U∞/δ. Let us define the Strouhal number
St = fL/U∞, where L is the separation length (L = 4.2δ in
the DNS). The range of StL is 0.03-0.042, which is consistent
with the range given by Dussauge et al. (2006).
Contours of the magnitude of the gradient of pressure
on streamwise-spanwise planes are plotted in Figure 3. Two
instantaneous flow fields are plotted at zn = 0.9δ and 2δ
away from the wall. At zn = 0.9, Figures 3(a) and (b),
the shock is nearly uniform in the spanwise direction. The
streamwise movement of the shock is roughly 1δ. Figures
3(c) and (d) plot the same times at a plane closer to the
wall. We observe a wrinkling of the shock in the spanwise
direction, with an amplitude of about 0.5δ. At zn = 0.9δ,
the shock also moves in the streamwise direction in the same
manner as shown in Figures 3(a) and (b). The amplitude of
the motion in the streamwise direction is twice that of the
spanwise wrinkling.
We analyze the shock motion within the context of these
two aspects. One is that the shock wrinkles along the span-
wise direction. The other corresponds to the larger ampli-
x/δ
y/δ
0 2 40
1
2 (a)
x/δ
y/δ
0 2 40
1
2 (b)
SKm SKsm
x/δ
y/δ
-2 0 20
1
2 (c)
x/δ
y/δ
-2 0 20
1
2 (d)
SKsmSKm
Figure 3: Contours of |∇p| showing the shock location for
two flow realizations separated by 50δ/U∞ at zn = 2δ ((a)
& (b)) and zn = 0.9δ ((c) & (d)). Dark indicates large
gradient. From Wu & Martin (2007b).
t (δ/U∞)
ρu/ρ
∞U
∞
0 100 200 3000.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
upstream of the shock (x=-2.9δ)inside shock motion region (x=0.8δ)downstream of the shock (x=1.5δ)
(a)
f (U∞/δ)
Eρu
fδ/ρ
2 U∞3
10-2 10-1
10-8
10-7
10-6
10-5
10-4
10-3
10-2
upstream of the shockinside shock motion regiondownstream of the shock
(b)
Figure 4: (a) Mass-flux signals and (b) corresponding pre-
multiplied energy spectra measured for different streamwise
locations at zn = 2δ. From Wu & Martin (2007).
tude motion upstream and downstream. The motion that is
inferred from the wall-pressure signal in Figure 2 results from
the combination of these two aspects. However, the low fre-
quency motion is related to the large-amplitude, streamwise
motion rather than to the spanwise wrinkling. This can be
seen from the mass-flux signals measured in the free stream
as shown in Figure 4. The signals are measured at different
streamwise locations (upstream, inside, and downstream of
the shock motion region) with a distance of 2δ away from the
wall along the centerline of the computational domain. In
Figure 4(a), the mass-flux signal measured inside the region
of shock motion oscillates between those measured upstream
and downstream, indicating that the shock is moving up-
stream and downstream of that point. The premultiplied
energy spectra plotted in Figure 4(b) show that the charac-
teristic low frequency range is between 0.007 − 0.013U∞/δ,
which is roughly the same as that given by the wall-pressure
signals in Figure 2(b).
Figure 5 plots normalized iso-surfaces of |∇ρ| for four
consecutive instantaneous flow fields. The structures in
the incoming boundary layer and the shock are seen. Two
structures are highlighted in Figures 5(a) and 5(c). For an
adiabatic wall, as in the DNS, these structures contain low-
density, low-speed fluid. As these structures pass through
the shock, the shock curves upstream, resulting in spanwise
wrinkling of the shock as shown in Figures 5(b) and (d).
From the data animation, the characteristic frequency of
spanwise wrinkling is O(U∞/δ).
To analyze the unsteadiness, we introduce two definitions
for the averaged shock location. First, the spanwise-mean
Figure 5: Iso-surface of |∇ρ| = 2ρ∞/δ showing structures
in the incoming boundary layer passing through the shock.
Temporal spacing between each frame is δ/U∞. From Wu
& Martin (2007b).
location, SKsm, in which the instantaneous location is de-
fined as the point where the pressure rises to 1.3p∞ in the
streamwise direction. Thus, SKsm is a function of time and
zn. Second, the absolute mean shock location, SKm, which
is computed by spanwise and temporal averaging the instan-
taneous shock location. In turn, SKm is only a function of
zn. Figure 3 (b) and (d) show SKsm and SKm locations.
The correlation with time lag between the pressure at
SKsm and the mass flux in the undisturbed incoming bound-
ary layer (5δ upstream of the ramp corner) is plotted in
Figure 6(a). Using SKsm in the correlation removes the
effect of the streamwise motion. The local correlation is
computed first using data on spanwise planes and then the
local correlations are spanwise averaged. The signals are
sampled at zn = 0.7δ since the shock is well defined there.
A peak of the correlation is observed at τ = −3.3δ/U∞
(i.e. events are separated about 3δ) with a magnitude of
about 0.35. The “enhanced” correlation is also plotted in
the same figure, where the contribution to the correlation
is only computed whenever the difference between the in-
stantaneous shock location and SKsm is greater than 0.15δ
(or 1.5 standard deviations). In other words, only strong
events are accounted for. The enhanced correlation has a
similar shape to the regular correlation. It peaks at the
same location with a greater magnitude, indicating that the
correlation is mainly influenced by strong events. Thus, the
spanwise wrinkling is related to low momentum fluid.
Figure 6(b) plots the correlation between pressure at the
absolute mean shock location, SKm, and the mass flux in
the undisturbed incoming boundary layer. For the regular
correlation, a peak is observed at the same location as in
Figure 6(a), but with a much smaller magnitude. The en-
hanced correlation is also computed, using data only when
the instantaneous shock location deviates from SKm more
than 0.3δ (or 1.5 standard deviations). Again, the enhanced
correlation peaks at the same location, however, the mag-
nitude observed is still much smaller than those in Figure
6(a). Measuring the mass flux of the incoming boundary
layer in the logarithmic region, where the superstructures are
best identified, gives equally low correlation values. Thus,
the streamwise shock motion is not significantly affected by
low momentum structures in the incoming boundary layer.
τ (δ/U∞)
corr
elat
ion
-6 -4 -2 0-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1 RegularEnhanced
(a)
τ (δ/U∞)
corr
elat
ion
-6 -4 -2 0-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
RegularEnhanced (b)
Figure 6: Spanwise-averaged correlation with time lag be-
tween (a) mass flux at (x = −5δ, y, zn = 0.7δ) and pressure
at (SKsm, y, zn = 0.7δ), and (b) mass flux at (x = −5δ, y,
zn = 0.7δ) and pressure at (SKm, y, zn = 0.7δ). From Wu
& Martin (2007b).
Computing the correlations in Figure 6 without spanwise
averaging gives the same result except that the correlation
curve is not as smooth due to the lesser number of samples.
Conditional statistics on the incoming boundary layer
have been calculated, conditionally based on the shock be-
ing upstream or downstream of the absolute mean location.
No significant difference is found in these properties. The
conditionally averaged mean profiles and boundary layer
parameters (Table 1) are nearly identical with very small
difference (consistently less than 3%). This is in agree-
ment with the experiments of Beresh et al. (2002) for a
28◦ compression ramp with M = 5, where the difference in
the conditionally averaged mean velocity was roughly 2%.
UNSTEADINESS OF THE SEPARATION BUBBLE
The separation and reattachment points (denoted by S
and R, respectively) are defined using a Cf = 0 criteria.
Figure 7 (a) plots the time evolution of the spanwise-mean
separation point Ssm and the reattachment point Rsm. The
spectra for these signals also exhibit a low frequency com-
ponent of about 0.01 U∞/δ. The shock foot is related to the
separation point because the flow turns first near the sepa-
ration bubble. Thus, we expect a strong correlation between
Ssm and SKsm. Figure 7 (b) plots the correlation for the
spanwise-mean separation point Ssm and SKsm at zn = 2δ.
The correlation peak is about 0.85 with a time lag of about
7δ/U∞. Notice the time interval between each data point
in Figure 7 (b) is about 3δ/U∞, therefore the peak loca-
tion has ±3δ/U∞ uncertainty. This uncertainty also applies
for all of the following correlations with time lag. Gana-
pathisubramani et al. (2007) correlated the instantaneous
separation point S (defined using a velocity threshold crite-
ria) and streamwise-averaged values of streamwise velocity
in the incoming boundary layer at zn = 0.2δ. The same anal-
ysis performed here yields a correlation of about 0.5, which
is similar to the value 0.4 found by Ganapathisubramani et
al. (2007). Figure 8a plots the profile for the correlation be-
tween the instantaneous separation point using the Cf = 0
definition and streamwise-averaged values of ρu, where the
streamwise averaging is performed from the separation point
to the inlet. Using the Cf = 0 criteria, the correlation factor
at zn = 0.2 is 0.23. Thus, the use of the actual definition of
the separation point decreases the correlation between the
separation point and the streamwise-averaged u significantly.
Figure 8b plots two correlations, the correlation between
Ssm and Rsm and the correlation between the shock location
SKsm and Rsm For the correlation between Ssm and Rsm,
t (δ/U∞)0 100 200 300
-4
-2
0
2
4
6Ssm
Rsm(a)
t (δ/U∞)
corr
elat
ion
-10 -5 0 5 10 15 200.2
0.4
0.6
0.8
1.0(b)
Figure 7: (a) Time evolution of the spanwise-mean separa-
tion and reattachment points and (b) correlation between
the spanwise-mean separation point Ssm and shock location
SKsm at zn = 2δ. From Wu & Martin (2007b).
zn+
corr
elat
ion
0 100 200
0.0
0.1
0.2
0.3
(a)
t (δ/U∞)
corr
elat
ion
-20 0 20-0.5
-0.4
-0.3
-0.2
-0.1
Ssm and Rsm
SKsm and Rsm
(b)
Figure 8: (a) Correlation profile between the instantaneous
separation point and streamwise averaged values of ρu and
(b) correlation between the separation and reattachment
point and the shock location at zn = 2 and the reattach-
ment point. From Wu & Martin (2007b).
a negative correlation is observed, indicating that the sep-
aration bubble undergoes a contraction/expansion motion.
Moreover, the peaks for both correlations are located at neg-
ative time lags, indicating that the motion of the separation
point (and the shock) lags that of the reattachment point.
This implies that the shock unsteadiness may be caused by
the flow inside the separation region, downstream of the
shock.
DISCUSSION
From the above analyses, the local spanwise wrinkling
shock motion is shown to correlate with low momentum fluid
in the incoming boundary layer, which is consistent with
what Wu & Miles (2001) found in a compression ramp inter-
action using high speed visualization techniques. However,
the spanwise wrinkling is a smaller scale, local unsteadiness
compared with the streamwise shock motion. The small cor-
relation between the low momentum fluid in the incoming
boundary layer and the separation point found in the DNS
implies that these low momentum structures might have a
relatively minor contribution to the shock unsteadiness. The
negative time lag in the correlation between the shock loca-
tion and reattachment point suggests that the separation re-
gion may play an important role in driving the low frequency
shock unsteadiness, as seen experimentally by Thomas et al.
(1994). The fact that the Strouhal number of the low fre-
quency shock motion defined using the separation length lies
in the experimental range (Dussauge et al. (2006)) is also
supportive for this argument. Pirozzoli & Grasso (2006) per-
formed a DNS of a reflected shock interaction and proposed
that the shock unsteadiness was sustained by an acoustic res-
onance mechanism that is responsible for generating tones
Figure 9: Streamlines in x − z planes showing break down
of the separation bubble. Time intervals are about 1δ/U∞.
From Wu & Martin (2007b).
in cavity flows. However, the low-frequency shock motion
may not be captured in their DNS due to the fact that the
lowest Strouhal number reported is between 0.09 and 0.24,
which is above the range 0.02-0.05 found in experiments.
According to Dussauge et al. (2006), the Strouhal number
of the low frequency motion does not seem to have a signifi-
cant dependence on Mach number, suggesting that acoustic
resonance may not cause the low-frequency shock motion.
It is interesting to point out that in cavity flows, there are
two modes observed (Gharib & Roshko, 1987; Rowley et al.,
2002): the shear-layer mode and the wake mode. In this
case, acoustic resonance is responsible for the generation of
the shear-layer mode, while the wake mode is purely hydro-
dynamic. Moreover, the wake mode corresponds to larger
scale and lower frequency motions compared to the shear-
layer mode. Providing that there are some similarities in
compression ramp interactions and cavity flows in that they
all have a shear layer formed above a separated region, we
suggest that the mechanism of the low-frequency shock un-
steadiness may resemble that of the generation of the wake
mode in cavity flows. In other separated flows, for example,
flow passing a backward-facing step, low frequency fluctua-
tions have also been indicated (e.g. Simpson, 1989), while
the driving mechanisms are still not fully understood.
DNS data animations show that the size (including the
length and height) of the separation bubble changes signifi-
cant with a low frequency that is comparable to that of the
low frequency shock motion. Figure 9 plots six consecutive
times in the DNS with time intervals of about δ/U∞, show-
ing the break down of the separation bubble indicated by
streamlines. Flow quantities are averaged in the spanwise
direction to get a clear picture. Contours of pressure gradi-
ent are also plotted to show the shock location. From frame
(c) to (f), fluid bursts outside the separation bubble, causing
the bubble to shrink. The shock then moves downstream at
a later time (not seen in the figure). To show how the sep-
aration bubble changes with time, the mass and the area of
reverse flow region inside the separation bubble are plotted
in Figure 10(a). The reverse flow region is defined as regions
in which u is negative, where u is spanwise averaged. It is
observed that the mass inside the reverse flow region has an
t (δ/U∞)0 100 200 300
0.0
0.1
0.2
0.3
0.4
0.5
0.6mass per unit span (ρ∞δ2)area (δ2)
(a)
t (δ/U∞)
corr
elat
ion
-20 -10 0 10 20-0.8
-0.6
-0.4
-0.2
0.0
0.2
SKsm
∆P
Figure 10: (a) Mass and volume of the reverse flow region
versus time and (b) Correlation between the mass inside the
reverse flow region with the spanwise mean shock location
SKsm at z = 2δ and with the wall pressure difference ∆P =
Pw(x = 1δ) − Pw(x = −2δ). From Wu & Martin (2007b).
intermittent character, just like the momentum signal inside
the shock motion region show in Figure 4. Figure 10(b) plots
the correlation of the mass signal with the spanwise averaged
shock location at z = 2δ. A high peak of 0.7 is observed at
about τ = −13δ/U∞ , showing the shock motion is closely
related to that of the separation bubble. In addition, the
shock motion lags that of the separation bubble, indicating
that the separation bubble drives the shock motion. Also,
Figure 10(b) plots the correlation of the mass signal with
the pressure difference between x = 1δ and x = −2δ, which
are close to the reattachment and separation points, respec-
tively. The pressure gradient decreases with increasing mass
of reverse flow, which is due to the enlargement of the sepa-
ration bubble in the streamwise direction and decreasing of
streamline curvature.
Based on the above observations, it is hypothesized that
one of the mechanisms driving the low-frequency shock mo-
tion can be described as a feedback loop between the sep-
aration bubble, the separated shear layer and the shock
system, which has some similarities with the cause of the
low-frequency “flapping motion” in backward-facing step
flows described by Eaton & Johnston (1981). That is, the
balance between shear layer entrainment from the separa-
tion bubble and injection near the reattachment point is
perturbed. If the injection is greater, the separation bubble
grows in size and causes the reattachment point to move
downstream and the separation point to move upstream.
The motion of the separation point causes the shock to move
with it. As the shock moves upstream, the pressure gradient
in the separation region decreases due to the enlargement of
the separation region and decreasing of streamline curvature.
The decreasing pressure gradient reduces the entrainment
of fluid into the separation bubble. In turn, the separa-
tion bubble becomes unstable and breaks down. When this
happens, fluid bursts outside the bubble and the separa-
tion region shrinks fairly rapidly, causing the shock to move
downstream at a later time. Similarly, when the shock moves
to a downstream location, the overall pressure gradient in
the separation region increases, which enhances entrainment
of fluid into the separation bubble, causing the bubble to
grow. Thus, the low frequency shock motion is closely re-
lated to the time scale associated with the growth and burst
of the separation bubble. Assuming that this time scale is
determined by the length of the separation bubble L and the
characteristic speed of the reverse flow UR, the dimension-
less shock frequency StR = fL/UR can be computed. Using
the maximum of the time-averaged reverse flow speed in the
separation bubble, 0.055U∞, to represent UR, the dimen-
sionless frequency StR in the DNS is around unity (about
0.8).
CONCLUSION
Wall-pressure and separation point signals indicate low-
frequency motions in DNS data of a 24◦ compression ramp.
Analyses show that the shock motion is characterized by
a low-frequency, large-amplitude streamwise motion with
characteristic frequency of about 0.013U∞/δ, and a rela-
tive smaller-amplitude, high-frequency O(U∞/δ) spanwise
wrinkling. The mass flux in the incoming boundary layer is
correlated with the high-frequency spanwise wrinkling mo-
tion. Conditional statistics indicate no significant difference
in the mean properties of the incoming boundary layer when
the shock is upstream/downstream.
The location of the separation point is highly correlated
with shock location with a time lag of about 7δ/U∞. A small
correlation is found between the low momentum structures
in the incoming boundary layer and the separation point,
indicating the influence of the superstructures on the shock
motion may be minor. However, it is found that both the
shock motion and the separation point motion are correlated
with and lag the motion of the reattachment point, suggest-
ing that the downstream flow plays an important role in
driving the low frequency shock motion. A model that is
described as a feedback loop between the separation bubble,
the separated shear layer, and the shock system is proposed
to explain the low frequency shock motion. Using the length
of the separation bubble and the characteristic reverse flow
speed (e.g. the maximum of the mean reverse flow speed),
the Strouhal number of the low frequency shock motion is
around unity.
We acknowledge insightful discussions with Prof. A.J.
Smits and the support from the Air Force Office of Scientific
Research under grant no. AF/9550-06-1-0323.
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